Symmetry in Crystallography: Understanding the International Tables

Paolo Radaelli

Abstract

This book provides a simple but rigorous introduction to crystallographic symmetry, up to the point where understanding and using the ‘International Tables for Crystallography’ (ITC) becomes possible. The ITC are a true ‘Bible’ for all fields of science dealing with matter in the crystalline state, from solid-state physics to chemistry to structural biology, and are frequently used (and sometimes abused) without a true understanding of the underlying concepts. The most essential set of topics required for a true understanding of the ITC are introduced gradually, with an emphasis on graphical r ... More

This book provides a simple but rigorous introduction to crystallographic symmetry, up to the point where understanding and using the ‘International Tables for Crystallography’ (ITC) becomes possible. The ITC are a true ‘Bible’ for all fields of science dealing with matter in the crystalline state, from solid-state physics to chemistry to structural biology, and are frequently used (and sometimes abused) without a true understanding of the underlying concepts. The most essential set of topics required for a true understanding of the ITC are introduced gradually, with an emphasis on graphical rather than algebraic notation (the graphical symbols used here are the same as in the ITC). This is done by first examining simple patterns, as found in architecture, art, graphic design, etc., and developing a graphical language to describe their symmetry and to combine symmetry operators. A minimal but reasonably rigorous set of group theoretical concepts, such as composition and the ‘multiplication table’ will be introduced to enable the manipulations of these symbols. As the dimensionality and complexity of the patterns increase from symmetry around a fixed point to quasi-1D (Frieze patterns) to 2D (plane or ‘wallpaper’ patterns), so will the set of required graphical symbols. Concepts such as site symmetry, multiplicity and special positions (denoted using Wyckoff letters), group-subgroup relations, etc., will also be introduced in a graphical way. Once the basic concepts are understood for these simple patterns, only a small conceptual leap is required to tackle the full set of the 230 space groups. In addition, the book contains sections on crystallographic coordinates and coordinate transformation, and on the symmetry of the so-called ‘reciprocal space’, which is essential to describing scattering experiments on crystals.

End Matter

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